Algebra Examples

${(2 c + 5 d)}^{3}$

Step 1

To write a polynomial in standard form, simplify and then arrange the terms in descending order.

$a x^{2} + b x + c$

Step 2

Use the Binomial Theorem.

${(2 c)}^{3} + 3 {(2 c)}^{2} (5 d) + 3 (2 c) {(5 d)}^{2} + {(5 d)}^{3}$

Step 3

Simplify each term.

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Step 3.1

Apply the product rule to $2 c$ .

$2^{3} c^{3} + 3 {(2 c)}^{2} (5 d) + 3 (2 c) {(5 d)}^{2} + {(5 d)}^{3}$

Step 3.2

Raise $2$ to the power of $3$ .

$8 c^{3} + 3 {(2 c)}^{2} (5 d) + 3 (2 c) {(5 d)}^{2} + {(5 d)}^{3}$

Step 3.3

Rewrite using the commutative property of multiplication.

$8 c^{3} + 3 \cdot 5 {(2 c)}^{2} d + 3 (2 c) {(5 d)}^{2} + {(5 d)}^{3}$

Step 3.4

Multiply $3$ by $5$ .

$8 c^{3} + 15 {(2 c)}^{2} d + 3 (2 c) {(5 d)}^{2} + {(5 d)}^{3}$

Step 3.5

Apply the product rule to $2 c$ .

$8 c^{3} + 15 (2^{2} c^{2}) d + 3 (2 c) {(5 d)}^{2} + {(5 d)}^{3}$

Step 3.6

Raise $2$ to the power of $2$ .

$8 c^{3} + 15 (4 c^{2}) d + 3 (2 c) {(5 d)}^{2} + {(5 d)}^{3}$

Step 3.7

Multiply $4$ by $15$ .

$8 c^{3} + 60 c^{2} d + 3 (2 c) {(5 d)}^{2} + {(5 d)}^{3}$

Step 3.8

Multiply $2$ by $3$ .

$8 c^{3} + 60 c^{2} d + 6 c {(5 d)}^{2} + {(5 d)}^{3}$

Step 3.9

Apply the product rule to $5 d$ .

$8 c^{3} + 60 c^{2} d + 6 c (5^{2} d^{2}) + {(5 d)}^{3}$

Step 3.10

Rewrite using the commutative property of multiplication.

$8 c^{3} + 60 c^{2} d + 6 \cdot 5^{2} c d^{2} + {(5 d)}^{3}$

Step 3.11

Raise $5$ to the power of $2$ .

$8 c^{3} + 60 c^{2} d + 6 \cdot 25 c d^{2} + {(5 d)}^{3}$

Step 3.12

Multiply $6$ by $25$ .

$8 c^{3} + 60 c^{2} d + 150 c d^{2} + {(5 d)}^{3}$

Step 3.13

Apply the product rule to $5 d$ .

$8 c^{3} + 60 c^{2} d + 150 c d^{2} + 5^{3} d^{3}$

Step 3.14

Raise $5$ to the power of $3$ .

$8 c^{3} + 60 c^{2} d + 150 c d^{2} + 125 d^{3}$